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The best beam steering directions are estimated through beam training, which is one of the most important and challenging tasks in millimeter-wave and sub-terahertz communications. Novel array architectures and signal processing techniques are required to avoid prohibitive beam training overhead associated with large antenna arrays and narrow beams. In this work, we leverage recent developments in true-time-delay (TTD) arrays with large delay-bandwidth products to accelerate beam training using frequency-dependent probing beams. We propose and study two TTD architecture candidates, including analog and hybrid analog-digital arrays, that can facilitate beam training with only one wideband pilot. We also propose a suitable algorithm that requires a single pilot to achieve high-accuracy estimation of angle of arrival. The proposed array architectures are compared in terms of beam training requirements and performance, robustness to practical hardware impairments, and power consumption. The findings suggest that the analog and hybrid TTD arrays achieve a sub-degree beam alignment precision with 66% and 25% lower power consumption than a fully digital array, respectively. Our results yield important design trade-offs among the basic system parameters, power consumption, and accuracy of angle of arrival estimation in fast TTD beam training.